Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}6x+9y &= -6 \\ 8x+3y &= 2\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}-6x-9y &= 6\\ 24x+9y &= 6\end{align*}$ Add the top and bottom equations. $18x = 12$ Divide both sides by $18$ and reduce as necessary. $x = \dfrac{2}{3}$ Substitute $\dfrac{2}{3}$ for $x$ in the top equation. $6( \dfrac{2}{3})+9y = -6$ $4+9y = -6$ $9y = -10$ $y = -\dfrac{10}{9}$ The solution is $\enspace x = \dfrac{2}{3}, \enspace y = -\dfrac{10}{9}$.